| import OAI.Analysis.FactorGeneration.ProductWords |
| import OAI.Analysis.FactorGeneration.ResidualCompletion |
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| namespace OAI |
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| noncomputable section |
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| namespace Generator.Tracial |
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| /-! Two-norm continuity of ordered moments along uniformly operator-bounded |
| representatives. No topological-ring structure on the two-norm quotient is used. -/ |
| section |
| open UltrapowerBridge Filter |
| open scoped Topology |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
|
|
| theorem quotient_bounded_mul_norm (a : Bounded M) (C : ℝ) (ha : ∀ j,‖a.val j‖ ≤ C) |
| (b : Q τ ω) : ‖quotientMap τ (ω : Filter ℕ) a*b‖ ≤ C*‖b‖ := by |
| induction b using Quotient.inductionOn with |
| | h b => |
| have hp := quotient_norm_tendsto τ ω (a*b) |
| have hb := (quotient_norm_tendsto τ ω b).const_mul C |
| rw [map_mul] at hp |
| apply le_of_tendsto_of_tendsto hp hb |
| exact Eventually.of_forall (fun j => (τ.norm₂_mul_le (a.val j) (b.val j)).trans |
| (mul_le_mul_of_nonneg_right (ha j) (norm_nonneg _))) |
|
|
| theorem quotient_mul_bounded_norm (a : Bounded M) (C : ℝ) (ha : ∀ j,‖a.val j‖ ≤ C) |
| (b : Q τ ω) : ‖b*quotientMap τ (ω : Filter ℕ) a‖ ≤ C*‖b‖ := by |
| induction b using Quotient.inductionOn with |
| | h b => |
| have hp := quotient_norm_tendsto τ ω (b*a) |
| have hb := (quotient_norm_tendsto τ ω b).const_mul C |
| rw [map_mul] at hp |
| apply le_of_tendsto_of_tendsto hp hb |
| apply Eventually.of_forall |
| intro j |
| calc τ.norm₂ (b.val j*a.val j) ≤ τ.norm₂ (b.val j)*‖a.val j‖ := τ.norm₂_mul_le_right _ _ |
| _ ≤ τ.norm₂ (b.val j)*C := mul_le_mul_of_nonneg_left (ha j) (norm_nonneg _) |
| _ = C*τ.norm₂ (b.val j) := mul_comm _ _ |
|
|
| theorem quotient_fixed_right_tendsto (b : Q τ ω) (a : ℕ → Q τ ω) (v : Q τ ω) |
| (ha : Tendsto a atTop (𝓝 v)) : Tendsto (fun n => a n*b) atTop (𝓝 (v*b)) := by |
| obtain ⟨b',hb⟩ := quotientMap_surjective τ (ω : Filter ℕ) b |
| obtain ⟨C,_,hbound⟩ := b'.property |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| have ht := (tendsto_iff_norm_sub_tendsto_zero.mp ha).const_mul C |
| apply squeeze_zero (fun n => norm_nonneg _) _ (by simpa only [mul_zero] using ht) |
| intro n |
| rw [← sub_mul,← hb] |
| exact quotient_mul_bounded_norm τ ω b' C hbound (a n-v) |
|
|
| theorem quotient_bounded_mul_tendsto (a : ℕ → Bounded M) (C : ℝ) |
| (hC : ∀ n j,‖(a n).val j‖ ≤ C) (v : Q τ ω) (b : ℕ → Q τ ω) (w : Q τ ω) |
| (ha : Tendsto (fun n => quotientMap τ (ω : Filter ℕ) (a n)) atTop (𝓝 v)) |
| (hb : Tendsto b atTop (𝓝 w)) : |
| Tendsto (fun n => quotientMap τ (ω : Filter ℕ) (a n)*b n) atTop (𝓝 (v*w)) := by |
| have ht1 := (tendsto_iff_norm_sub_tendsto_zero.mp hb).const_mul C |
| have ht2 := tendsto_iff_norm_sub_tendsto_zero.mp (quotient_fixed_right_tendsto τ ω w _ v ha) |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| apply squeeze_zero (fun n => norm_nonneg _) _ (by simpa only [mul_zero,zero_add] using ht1.add ht2) |
| intro n |
| have he : quotientMap τ (ω : Filter ℕ) (a n)*b n-v*w= |
| quotientMap τ (ω : Filter ℕ) (a n)*(b n-w)+ |
| (quotientMap τ (ω : Filter ℕ) (a n)*w-v*w) := by rw [mul_sub,sub_add_sub_cancel] |
| rw [he] |
| exact (norm_add_le (quotientMap τ (ω : Filter ℕ) (a n)*(b n-w)) |
| (quotientMap τ (ω : Filter ℕ) (a n)*w-v*w)).trans |
| (add_le_add (quotient_bounded_mul_norm τ ω (a n) C (hC n) (b n-w)) le_rfl) |
|
|
| theorem quotient_fixed_left_tendsto (b : Q τ ω) (a : ℕ → Q τ ω) (v : Q τ ω) |
| (ha : Tendsto a atTop (𝓝 v)) : Tendsto (fun n => b*a n) atTop (𝓝 (b*v)) := by |
| obtain ⟨b',rfl⟩ := quotientMap_surjective τ (ω : Filter ℕ) b |
| obtain ⟨C,_,hbound⟩ := b'.property |
| exact quotient_bounded_mul_tendsto τ ω (fun _ => b') C (fun _ => hbound) _ a v tendsto_const_nhds ha |
|
|
| theorem quotient_star_tendsto (a : ℕ → Q τ ω) (v : Q τ ω) |
| (ha : Tendsto a atTop (𝓝 v)) : Tendsto (fun n => star (a n)) atTop (𝓝 (star v)) := by |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| simpa only [← star_sub,quotient_norm_star] using tendsto_iff_norm_sub_tendsto_zero.mp ha |
|
|
| theorem quotient_partial_signed_tendsto (a : ℕ → ℕ → PartialUnitary M) (v : Q τ ω) |
| (ha : Tendsto (fun n => quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) atTop (𝓝 v)) |
| (s : Bool) : Tendsto (fun n => quotientMap τ (ω : Filter ℕ) |
| (algebraSignedLetter (boundedPartialUnitary (a n)) s)) atTop (𝓝 (algebraSignedLetter v s)) := by |
| cases s with |
| | false => exact ha |
| | true => simpa only [algebraSignedLetter,↓reduceIte,quotientMap_star] using quotient_star_tendsto τ ω _ v ha |
|
|
| theorem quotient_partial_core_tendsto (a : ℕ → ℕ → PartialUnitary M) (v : Q τ ω) |
| (ha : Tendsto (fun n => quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) atTop (𝓝 v)) |
| (x : ℕ → Q τ ω) (s : Bool) (xs : List (Bool × ℕ)) : |
| Tendsto (fun n => algebraWordCore (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) x s xs) |
| atTop (𝓝 (algebraWordCore v x s xs)) := by |
| have hsign (t : Bool) := quotient_partial_signed_tendsto τ ω a v ha t |
| have he (n : ℕ) (t : Bool) : quotientMap τ (ω : Filter ℕ) |
| (algebraSignedLetter (boundedPartialUnitary (a n)) t)= |
| algebraSignedLetter (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) t := by |
| cases t <;> simp only [algebraSignedLetter,Bool.false_eq_true,↓reduceIte,quotientMap_star] |
| induction xs with |
| | nil => simpa only [algebraWordCore,he] using hsign s |
| | cons t xs ih => |
| have hbound : ∀ n j,‖(algebraSignedLetter (boundedPartialUnitary (a n)) t.1).val j‖ ≤ 1 := by |
| intro n j |
| cases t.1 with |
| | false => |
| change ‖(a n j).val‖ ≤ 1 |
| exact (a n j).unitary.norm_le_one |
| | true => |
| change ‖star (a n j).val‖ ≤ 1 |
| exact (a n j).unitary.adjoint.norm_le_one |
| have hh := quotient_bounded_mul_tendsto τ ω |
| (fun n => algebraSignedLetter (boundedPartialUnitary (a n)) t.1) 1 hbound _ |
| (fun n => x t.2*algebraWordCore (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) x s xs) |
| _ (hsign t.1) (quotient_fixed_left_tendsto τ ω (x t.2) _ _ ih) |
| simpa only [he,algebraWordCore,mul_assoc] using hh |
|
|
| theorem quotient_trace_tendsto (a : ℕ → Q τ ω) (v : Q τ ω) |
| (ha : Tendsto a atTop (𝓝 v)) : |
| Tendsto (fun n => quotientTrace τ ω (a n)) atTop (𝓝 (quotientTrace τ ω v)) := by |
| have he (b : Q τ ω) : quotientTrace τ ω b=inner ℂ (1 : Q τ ω) b := by |
| rw [inner_eq_trace,star_one,one_mul] |
| simp only [he] |
| exact tendsto_const_nhds.inner ha |
|
|
| theorem exactOrderedMoments_partial_limit (a : ℕ → ℕ → PartialUnitary M) (v : Q τ ω) |
| (ha : Tendsto (fun n => quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) atTop (𝓝 v)) |
| (x : ℕ → Q τ ω) |
| (hm : ∀ n,ExactOrderedMoments τ ω |
| (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary (a n))) x) : |
| ExactOrderedMoments τ ω v x := by |
| intro first last xs s |
| have ht := quotient_partial_core_tendsto τ ω a v ha x s xs |
| have ht' := quotient_trace_tendsto τ ω _ _ (quotient_fixed_left_tendsto τ ω |
| (optionalCoefficient x first) _ _ (quotient_fixed_right_tendsto τ ω (optionalCoefficient x last) _ _ ht)) |
| rw [algebraWordCore_endpoints] |
| apply tendsto_nhds_unique ht' |
| apply tendsto_const_nhds.congr |
| intro n |
| have hh := hm n first last xs s |
| rw [algebraWordCore_endpoints] at hh |
| exact hh.symm |
|
|
| end |
|
|
| /-! A genuine unitary in the included quotient with every signed ordered |
| moment zero. This statement does not assert all Haar powers or freeness. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
|
|
| theorem exactOrderedMoments_zero (ω : Ultrafilter ℕ) (x : ℕ → Q τ ω) : |
| ExactOrderedMoments τ ω 0 x := by |
| intro first last xs s |
| have hz : algebraPatchingWord (0 : Q τ ω) |
| (xs.map (fun a => (a.1,x a.2)) ++ [(s,optionalCoefficient x last)])=0 := by |
| cases xs <;> simp [algebraPatchingWord,algebraSignedLetter] |
| rw [hz,mul_zero] |
| exact (traceLinear τ ω).map_zero |
|
|
| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
|
|
| theorem exists_full_support_ordered_independence |
| (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop) |
| (x : ℕ → Q τ ω) (hx : ∀ i,quotientTrace τ ω (x i)=0) : |
| ∃ u : ℕ → PartialUnitary M, |
| (∀ j,(u j).val.val ∈ N) ∧ |
| star (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u))* |
| quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u)=1 ∧ |
| quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u)* |
| star (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u))=1 ∧ |
| ExactOrderedMoments τ ω (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u)) x := by |
| let v : ℕ → PartialUnitary M := fun _ => zeroPartialUnitary |
| have hv : ∀ j,(v j).val.val ∈ N := fun _ => N.zero_mem |
| have hz : quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary v)=0 := by |
| change quotientMap τ (ω : Filter ℕ) 0=0 |
| exact map_zero _ |
| have hm : ExactOrderedMoments τ ω (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary v)) x := by |
| rw [hz] |
| exact exactOrderedMoments_zero τ ω x |
| obtain ⟨a,ha⟩ := exists_exact_residual_iteration (H := H) (N := N) (M := M) |
| τ hNM hfactor hdiff hirr hsep ω hω v hv x hx hm |
| obtain ⟨u,hu⟩ := exists_coordinate_monotone_upper τ hNM a ha.2.2.1 ha.2.1 |
| have hc := quotient_geometric_upper_completion τ ω a u hu.2 |
| (quotientResidualTrace τ ω v) ha.2.2.2.2 |
| have hunit := quotient_unitary_of_zero_residual τ ω u hc.1 |
| exact ⟨u,hu.1,hunit.1,hunit.2, |
| exactOrderedMoments_partial_limit τ ω a _ hc.2 x ha.2.2.2.1⟩ |
|
|
| / |
| theorem exists_included_ordered_unitary |
| (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop) |
| (x : ℕ → Q τ ω) (hx : ∀ i,quotientTrace τ ω (x i)=0) : |
| ∃ t : Q τ ω, |
| (∃ q : TracialQuotient (restrictedTrace τ hNM) (ω : Filter ℕ), |
| quotientInclusion τ hNM (ω : Filter ℕ) q=t) ∧ |
| star t*t=1 ∧ t*star t=1 ∧ ExactOrderedMoments τ ω t x := by |
| obtain ⟨u,hu,hi,hf,hm⟩ := exists_full_support_ordered_independence τ hNM hfactor hdiff hirr hsep |
| ω hω x hx |
| obtain ⟨b,hb⟩ := boundedPartialUnitary_inclusion hNM u hu |
| refine ⟨quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u),?_,hi,hf,hm⟩ |
| refine ⟨quotientMap (restrictedTrace τ hNM) (ω : Filter ℕ) b,?_⟩ |
| change quotientMap τ (ω : Filter ℕ) (boundedInclusion hNM b)=_ |
| rw [hb] |
|
|
| end |
|
|
| /-! Two actual applications of ordered independence produce a unitary with |
| all reduced signed moments, including runs of same-sign letters. -/ |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
|
|
| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
|
|
| theorem exists_included_reduced_unitary |
| (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop) |
| (x : ℕ → Q τ ω) (hx : ∀ i,quotientTrace τ ω (x i)=0) : |
| ∃ t : unitary (Q τ ω), |
| (∃ q : TracialQuotient (restrictedTrace τ hNM) (ω : Filter ℕ), |
| quotientInclusion τ hNM (ω : Filter ℕ) q=(t : Q τ ω)) ∧ |
| ReducedOrderedMoments τ ω (t : Q τ ω) x := by |
| obtain ⟨u,⟨qu,hqu⟩,hui,huf,hum⟩ := |
| exists_included_ordered_unitary τ hNM hfactor hdiff hirr hsep ω hω x hx |
| obtain ⟨v,⟨qv,hqv⟩,hvi,hvf,hvm⟩ := |
| exists_included_ordered_unitary τ hNM hfactor hdiff hirr hsep ω hω |
| (productCenteredFamily τ ω u x) (productCenteredFamily_centered τ ω u x huf hx hum) |
| have hunit : u*v ∈ unitary (Q τ ω) := by |
| apply (unitary (Q τ ω)).mul_mem |
| · exact ⟨hui,huf⟩ |
| · exact ⟨hvi,hvf⟩ |
| refine ⟨⟨u*v,hunit⟩,⟨qu*qv,?_⟩,?_⟩ |
| · rw [map_mul,hqu,hqv] |
| · intro a b s xs hxs |
| exact product_reduced_moment τ ω u v x hvm a b s xs hxs |
|
|
| end Generator.Tracial |
|
|
| end |
|
|
| end OAI |
|
|
